This is the first comprehensive introduction to the powerful moment approach for solving global optimization problems (and some related problems) described by polynomials (and even semi-algebraic functions). In particular, the author explains how to use relatively recent results from real algebraic geometry to provide a systematic numerical scheme for computing the optimal value and global minimizers. Indeed, among other things, powerful positivity certificates from real algebraic geometry allow one to define an appropriate hierarchy of semidefinite (SOS) relaxations or LP relaxations whose optimal values converge to the global minimum. Several extensions to related optimization problems are also described. Graduate students, engineers and researchers entering the field can use this book to understand, experiment with and master this new approach through the simple worked examples provided.

Why do some games seem to be universal while others have a particular connection to the culture of the people playing them? Around the World in 80 Games is about the mathematics of chance, game theory, gamification, gaming strategies and computer games. Traversing the globe, Marcus du Sautoy looks at the genesis of games new and old, explores how to invent a good game and explains the fascination of a popular lockdown game. From the secrets of whist to hopscotch, Scrabble to Wordle. The most simple games endure: board games, card games and dice games have captivated us for centuries and the acclaimed mathematician and author of The Creativity Code (among many others) will once again bring mathematics to the fore with insight and aplomb in Around the World in 80 Games.

Many important applications in global optimization, algebra, probability and statistics, applied mathematics, control theory, financial mathematics, inverse problems, etc. can be modeled as a particular instance of the Generalized Moment Problem (GMP).This book introduces a new general methodology to solve the GMP when its data are polynomials and basic semi-algebraic sets. This methodology combines semidefinite programming with recent results from real algebraic geometry to provide a hierarchy of semidefinite relaxations converging to the desired optimal value. Applied on appropriate cones, standard duality in convex optimization nicely expresses the duality between moments and positive polynomials.In the second part, the methodology is particularized and described in detail for various applications, including global optimization, probability, optimal control, mathematical finance, multivariate integration, etc., and examples are provided for each particular application.

"Mathematical Optimization and Economic Analysis" is a self-contained introduction to various optimization techniques used in economic modeling and analysis such as geometric, linear, and convex programming and data envelopment analysis. Through a systematic approach, this book demonstrates the usefulness of these mathematical tools in quantitative and qualitative economic analysis.

The book presents specific examples to demonstrate each technique's advantages and applicability as well as numerous applications of these techniques to industrial economics, regulatory economics, trade policy, economic sustainability, production planning, and environmental policy.

Key Features include:

- A detailed presentation of both single-objective and multiobjective optimization;

- An in-depth exposition of various applied optimization problems;

- Implementation of optimization tools to improve the accuracy of various economic models;

- Extensive resources suggested for further reading.

This book is intended for graduate and postgraduate students studying quantitative economics, as well as economics researchers and applied mathematicians. Requirements include a basic knowledge of calculus and linear algebra, and a familiarity with economic modeling.

Steadily growing applications of game theory in modern science (including psychology, biology and economics) require sources to provide rapid access in both classical tools and recent developments to readers with diverse backgrounds. This book on game theory, its applications and mathematical methods, is written with this objective in mind.The book gives a concise but wide-ranging introduction to games including older (pre-game theory) party games and more recent topics like elections and evolutionary games and is generously spiced with excursions into philosophy, history, literature and politics. A distinguished feature is the clear separation of the text into two parts: elementary and advanced, which makes the book ideal for study at various levels.Part I displays basic ideas using no more than four arithmetic operations and requiring from the reader only some inclination to logical thinking. It can be used in a university degree course without any (or minimal) prerequisite in mathematics (say, in economics, business, systems biology), as well as for self-study by school teachers, social and natural scientists, businessmen or laymen.Part II is a rapid introduction to the mathematical methods of game theory, suitable for a mathematics degree course of various levels. It includes an advanced material not yet reflected in standard textbooks, providing links with the exciting modern developments in financial mathematics (rainbow option pricing), tropical mathematics, statistical physics (interacting particles) and discusses structural stability, multi-criteria differential games and turnpikes.To stimulate the mathematical and scientific imagination, graphics by a world-renowned mathematician and mathematics imaging artist, A T Fomenko, are used. The carefully selected works of this artist fit remarkably into the many ideas expressed in the book.

Herbert Scarf is a highly esteemed distinguished American economist. He is internationally famous for his early epoch-making work on optimal inventory policies and his highly influential study with Andrew Clark on optimal policies for a multi-echelon inventory problem, which initiated the important and flourishing field of supply chain management. Equally, he has gained world recognition for his classic study on the stability of the Walrasian price adjustment processes and his fundamental analysis on the relationship between the core and the set of competitive equilibria (the so-called Edgeworth conjecture). Further achievements include his remarkable sufficient condition for the existence of a core in non-transferable utility games and general exchange economies, his seminal paper with Lloyd Shapley on housing markets, and his pioneering study on increasing returns and models of production in the presence of indivisibilities. All in all, however, the name of Scarf is always remembered as a synonym for the computation of economic equilibria and fixed points. In the early 1960s he invented a path-breaking technique for computing equilibrium prices.This work has generated a major research field in economics termed Applied General Equilibrium Analysis and a corresponding area in operations research known as Simplicial Fixed Point Methods. This book comprises all his research articles and consists of four volumes. The volume collects Herbert Scarfs papers in the area of Production in Indivisibilities and the Theories of Large Firms.

This book offers a rigorous mathematical analysis of fuzzy geometrical ideas. It demonstrates the use of fuzzy points for interpreting an imprecise location and for representing an imprecise line by a fuzzy line. Further, it shows that a fuzzy circle can be used to represent a circle when its description is not known precisely, and that fuzzy conic sections can be used to describe imprecise conic sections. Moreover, it discusses fundamental notions on fuzzy geometry, including the concepts of fuzzy line segment and fuzzy distance, as well as key fuzzy operations, and includes several diagrams and numerical illustrations to make the topic more understandable. The book fills an important gap in the literature, providing the first comprehensive reference guide on the fuzzy mathematics of imprecise image subsets and imprecise geometrical objects. Mainly intended for researchers active in fuzzy optimization, it also includes chapters relevant for those working on fuzzy image processing and pattern recognition. Furthermore, it is a valuable resource for beginners interested in basic operations on fuzzy numbers, and can be used in university courses on fuzzy geometry, dealing with imprecise locations, imprecise lines, imprecise circles, and imprecise conic sections.

Games provide mathematical models for interaction. Numerous tasks in computer science can be formulated in game-theoretic terms. This fresh and intuitive way of thinking through complex issues reveals underlying algorithmic questions and clarifies the relationships between different domains. This collection of lectures, by specialists in the field, provides an excellent introduction to various aspects of game theory relevant for applications in computer science that concern program design, synthesis, verification, testing and design of multi-agent or distributed systems. Originally devised for a Spring School organised by the GAMES Networking Programme in 2009, these lectures have since been revised and expanded, and range from tutorials concerning fundamental notions and methods to more advanced presentations of current research topics. This volume is a valuable guide to current research on game-based methods in computer science for undergraduate and graduate students. It will also interest researchers working in mathematical logic, computer science and game theory.

This self-contained textbook discusses all major topics in functional analysis. Combining classical materials with new methods, it supplies numerous relevant solved examples and problems and discusses the applications of functional analysis in diverse fields. The book is unique in its scope, and a variety of applications of functional analysis and operator-theoretic methods are devoted to each area of application. Each chapter includes a set of problems, some of which are routine and elementary, and some of which are more advanced. The book is primarily intended as a textbook for graduate and advanced undergraduate students in applied mathematics and engineering. It offers several attractive features making it ideally suited for courses on functional analysis intended to provide a basic introduction to the subject and the impact of functional analysis on applied and computational mathematics, nonlinear functional analysis and optimization. It introduces emerging topics like wavelets, Gabor system, inverse problems and application to signal and image processing.

This book develops the concepts of various unique optimization techniques in the crisp and fuzzy environment. It provides an extensive overview of geometric programming methods within a unifying framework, and presents an in-depth discussion of the modified geometric programming problem, fuzzy geometric programming, as well as new insights into goal geometric programming. With numerous examples and exercises together with detailed solutions for several problems, the book also addresses fuzzy multi-objective geometric programming techniques. Geometric programming, which falls into the general class of signomial problems, has applications across disciplines, from engineering to economics, and is extremely useful in applications of a variety of optimization problems. Organized into thirteen chapters, this book is a valuable resource for graduate and advanced undergraduate students and researchers in applied mathematics and engineering.

Nonnegative matrix factorization (NMF) in its modern form has become a standard tool in the analysis of high-dimensional data sets. This book provides a comprehensive and up-to-date account of the most important aspects of the NMF problem and is the first to detail its theoretical aspects, including geometric interpretation, nonnegative rank, complexity, and uniqueness. It explains why understanding these theoretical insights is key to using this computational tool effectively and meaningfully. Nonnegative Matrix Factorization is accessible to a wide audience and is ideal for anyone interested in the workings of NMF. It discusses some new results on the nonnegative rank and the identifiability of NMF and makes available MATLAB codes for readers to run the numerical examples presented in the book. Graduate students starting to work on NMF and researchers interested in better understanding the NMF problem and how they can use it will find this book useful. It can be used in advanced undergraduate and graduate-level courses on numerical linear algebra and on advanced topics in numerical linear algebra and requires only a basic knowledge of linear algebra and optimization.

Mathematical Optimization Terminology: A Comprehensive Glossary of Terms is a practical book with the essential formulations, illustrative examples, real-world applications and main references on the topic. This book helps readers gain a more practical understanding of optimization, enabling them to apply it to their algorithms. This book also addresses the need for a practical publication that introduces these concepts and techniques.

Although valued for its ability to allow teams to collaborate and foster coalitional behaviors among the participants, game theory's application to networking systems is not without challenges. Distributed Strategic Learning for Wireless Engineers illuminates the promise of learning in dynamic games as a tool for analyzing network evolution and underlines the potential pitfalls and difficulties likely to be encountered. Establishing the link between several theories, this book demonstrates what is needed to learn strategic interaction in wireless networks under uncertainty, randomness, and time delays. It addresses questions such as: How much information is enough for effective distributed decision making? Is having more information always useful in terms of system performance? What are the individual learning performance bounds under outdated and imperfect measurement? What are the possible dynamics and outcomes if the players adopt different learning patterns? If convergence occurs, what is the convergence time of heterogeneous learning? What are the issues of hybrid learning? How can one develop fast and efficient learning schemes in scenarios where some players have more information than the others? What is the impact of risk-sensitivity in strategic learning systems? How can one construct learning schemes in a dynamic environment in which one of the players do not observe a numerical value of its own-payoffs but only a signal of it? How can one learn "unstable" equilibria and global optima in a fully distributed manner? The book provides an explicit description of how players attempt to learn over time about the game and about the behavior of others. It focuses on finite and infinite systems, where the interplay among the individual adjustments undertaken by the different players generates different learning dynamics, heterogeneous learning, risk-sensitive learning, and hybrid dynamics.

Herbert Scarf is a highly esteemed distinguished American economist. He is internationally famous for his early epoch-making work on optimal inventory policies and his highly influential study with Andrew Clark on optimal policies for a multi-echelon inventory problem, which initiated the important and flourishing field of supply chain management. Equally, he has gained world recognition for his classic study on the stability of the Walrasian price adjustment processes and his fundamental analysis on the relationship between the core and the set of competitive equilibria (the so-called Edgeworth conjecture). Further achievements include his remarkable sufficient condition for the existence of a core in non-transferable utility games and general exchange economies, his seminal paper with Lloyd Shapley on housing markets, and his pioneering study on increasing returns and models of production in the presence of indivisibilities. All in all, however, the name of Scarf is always remembered as a synonym for the computation of economic equilibria and fixed points. In the early 1960s he invented a path-breaking technique for computing equilibrium prices.This work has generated a major research field in economics termed Applied General Equilibrium Analysis and a corresponding area in operations research known as Simplicial Fixed Point Methods. This book comprises all his research articles and consists of four volumes. This volume collects Herbert Scarf's papers in the area of Operations Research and Management.

Introduction to Nature-Inspired Optimization brings together many of the innovative mathematical methods for non-linear optimization that have their origins in the way various species behave in order to optimize their chances of survival. The book describes each method, examines their strengths and weaknesses, and where appropriate, provides the MATLAB code to give practical insight into the detailed structure of these methods and how they work. Nature-inspired algorithms emulate processes that are found in the natural world, spurring interest for optimization. Lindfield/Penny provide concise coverage to all the major algorithms, including genetic algorithms, artificial bee colony algorithms, ant colony optimization and the cuckoo search algorithm, among others. This book provides a quick reference to practicing engineers, researchers and graduate students who work in the field of optimization.

This book serves as an introduction to game theory for students with no prior game theory knowledge, or with limited background in economics and mathematics. It is specifically designed to provide an intuitive and accessible interdisciplinary approach to game theory, while simultaneously exploring cooperative games, repeated play, correlated equilibrium, and a range of applications.

This textbook offers a comprehensive overview of the main developments in game theory since the 1950s. It provides a wide variety of examples and exercises, mostly drawn from applications in economics, to illustrate key concepts and ideas in the field. The book should prove an invaluable reference tool for teachers, students, and researchers of microeconomics and game theory.

This is a series of interdisciplinary conferences in mathematics, philosophy, computer science and linguistics. The main goal is to re-establish the traditionally strong links between these areas of research that have been lost in the past decades. The second conference in the series had the subtitle Applications of Mathematical Logic in Philosophy and Linguistics and brought speakers from all parts of the formal sciences together to give a holistic view of how mathematical methods can improve our philosophical and technical understanding of language and scientific discourse, ranging from the theoretical level up to applications in language recognition software.

The structures of the world's national and international, political and economic institutions have largely resulted from intuitive and ad hoc organization with reforms taking place on a costly trial-and-error basis. This book offers a comprehensive evaluation of the tools which can be used for a more rational and formal approach to institutional design. As new institutions and, indeed, new national governments are being formed and developed all the time, there is a considerable need for formal models that can facilitate their design. This book offers conceptual approaches and theories that shed new light on how various social and political institutions can emerge as the outcome of goal-directed rational behaviour. The author provides tools for evaluating existing institutions and for setting up new ones, demonstrating the applicability of decision and game theoretic tools, social choice theory and mechanism design to the construction of political and economic institutions. Using these approaches he particularly discusses the practical implications for the design of institutions in the European Union. This important book will be welcomed by students and scholars interested in government and political science, rational choice theory, methodology in the social sciences, and the microeconomics of rational behaviour.

Mathematical Puzzle Tales from Mount Olympus uses fascinating tales from Greek Mythology as the background for introducing mathematics puzzles to the general public. A background in high school mathematics will be ample preparation for using this book, and it should appeal to anyone who enjoys puzzles and recreational mathematics. Features: Combines the arts and science, and emphasizes the fact that mathematics straddles both domains. Great resource for students preparing for mathematics competitions, and the trainers of such students.

Game Theory: A Modeling Approach quickly moves readers through the fundamental ideas of the subject to enable them to engage in creative modeling projects based on game theoretic concepts. The authors match conclusions to real-world scenarios and applications. The text engages students in active learning, group work, in-class discussions and interactive simulations. Each chapter provides foundation pieces or adds more features to help readers build game theoretic models. The chapters include definitions, concepts and illustrative examples. The text will engage and challenge both undergraduate and graduate students. Features: Enables readers to apply game theorty to real-world scenarios Chapters can be used for core course materials or independent stuides Exercises, included at the end of the chapters, follow the order of the sections in the text Select answers and solutions are found at the end of the book Solutions manual for instructors is available from the authors

Predictive Control is aimed at students wishing to learn predictive control, as well as teachers, engineers and technicians of the profession. The book proposes a simple predictive controller where the control laws are given in clear text that requires no calculations. Adjustment, reduced to one or two parameters, is particularly easy, by means of charts, thus allowing the operator to choose the horizon according to the desired performances. Implementation is discussed in detail in two forms: RS or RST controller in z-1, and pseudo-code realization algorithms for a complete program (model and controller). The book is simple and practical, with the aim of the industrial implementation of many processes: Broida models, Strejc, integrators, dual integrators, with delay, or with inverse response. All settings are abundantly illustrated with response curves.

This book presents an introduction to variational analysis, a field which unifies theories and techniques developed in calculus of variations, optimization, and control, and covers convex analysis, nonsmooth analysis, and set-valued analysis. It focuses on problems with constraints, the analysis of which involves set-valued mappings and functions that are not differentiable. Applications of variational analysis are interdisciplinary, ranging from financial planning to steering a flying object. The book is addressed to graduate students, researchers, and practitioners in mathematical sciences, engineering, economics, and finance. A typical reader of the book should be familiar with multivariable calculus and linear algebra. Some basic knowledge in optimization, control, and elementary functional analysis is desirable, but all necessary background material is included in the book.

Multidimensional continued fractions form an area of research within number theory. Recently the topic has been linked to research in dynamical systems, and mathematical physics, which means that some of the results discovered in this area have applications in describing physical systems. This book gives a comprehensive and up to date overview of recent research in the area.

This book provides readers with a comprehensive survey of models of dynamic games in economics, including an extensive coverage of numerous fields of applications. It will also discuss and explain main concepts and techniques used in dynamic games, and inform readers of its major developments while equipping them with tools and ideas that will aid in the formulation of solutions for problems. A Survey of Dynamic Games in Economics will interest those who wish to study more about the conceptions, approaches and models that are applied in the domain of dynamic games.